Problem: Simplify and expand the following expression: $ \dfrac{k - 2}{5k - 10}-\dfrac{4k + 1}{3k + 1} $
Solution: In order to subtract expressions, they must have a common denominator. Get both fractions over a common denominator of $(5k - 10)(3k + 1)$ Multiply the first term by $\dfrac{3k + 1}{3k + 1}$ $ \begin{align*} \dfrac{k - 2}{5k - 10} \times \dfrac{3k + 1}{3k + 1} & = \dfrac{(k - 2)(3k + 1)}{(5k - 10)(3k + 1)} \\ & = \dfrac{3k^2 - 5k - 2}{(5k - 10)(3k + 1)}\end{align*} $ Multiply the second term by $\dfrac{5k - 10}{5k - 10}$ $ \begin{align*} \dfrac{4k + 1}{3k + 1} \times \dfrac{5k - 10}{5k - 10} & = \dfrac{(4k + 1)(5k - 10)}{(3k + 1)(5k - 10)} \\ & = \dfrac{20k^2 - 35k - 10}{(3k + 1)(5k - 10)}\end{align*} $ Now we have: $ = \dfrac{3k^2 - 5k - 2}{(5k - 10)(3k + 1)} - \dfrac{20k^2 - 35k - 10}{(3k + 1)(5k - 10)} $ Now both terms have a common denominator we can subtract the numerators: $ = \dfrac{3k^2 - 5k - 2 - (20k^2 - 35k - 10)}{(5k - 10)(3k + 1)} $ $ = \dfrac{3k^2 - 5k - 2 - 20k^2 + 35k + 10}{(5k - 10)(3k + 1)} $ $ = \dfrac{-17k^2 + 30k + 8}{(5k - 10)(3k + 1)}$ Expand the denominator: $ = \dfrac{-17k^2 + 30k + 8}{15k^2 - 25k - 10}$